⊗Quantum Fourier transform

Fourier analysis of discrete amplitudes.

1. Core idea

The QFT changes basis so periodic phase structure becomes concentrated in measurable frequency labels. Standard circuit constructions use Hadamard and controlled-phase rotations, plus optional final swaps to reverse bit order. QFT alone does not expose an entire spectrum in one shot.

2. Mathematical foundation

The notation describes a precise relationship among states, transformations or probabilities. Identify each symbol, check the dimensions of the objects, and distinguish amplitudes from the probabilities produced by a measurement.

3. Worked example

4. Reasoning flow

1. Define the initial register and its basis states.
2. Apply the transformations in their intended order.
3. Track amplitude magnitudes and relative phases.
4. Measure in the basis that reveals the desired property.
5. Interpret the result and check the resource assumptions.

Qiskit implementation

Run this standalone Python script with Qiskit installed. It uses exact statevector simulation rather than noisy hardware shots.

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector
from qiskit.circuit.library import QFTGate
qc = QuantumCircuit(2)
qc.x(0)                   # prepare basis state |01> (Qiskit bit order)
qc.append(QFTGate(2), [0, 1])  # discrete Fourier transform
print(qc.draw())
print(Statevector.from_instruction(qc))
# Four amplitudes of magnitude 1/2, with differing phases.

5. Check your understanding

6. What to watch out for

Ideal-state calculations assume perfect gates and measurements. When assessing an algorithm, separate its mathematical guarantee from the cost of building oracles, preparing inputs, repeating experiments and correcting physical noise.